DEMAND SYSTEMS AND FREQUENCY OF PURCHASE MODELS* Arnar Buason,1 Dadi Kristofersson1 and Kyrre Rickertsen2
* The authors thank Pierre Combris and INRA, Paris for help with obtaining and
preparing the data. The Research Council of Norway (Grant 199564) provided financial support for this research.
1 Department of Economics, University of Iceland, Reykjavik, Iceland. Corresponding author is Arnar Buason [email protected]
2 School of Economics and Business, Norwegian University of Life Sciences, P.O. Box 5003, 1432 Ås, Norway.
2 ABSTRACT
Understanding the frequencies of purchase and the average purchased quantities is important for marketing strategies such as the loss-leader pricing strategy. We develop a microeconomic model where the total purchases of goods are determined by demand systems for the frequencies of purchases and the average quantities purchased, conditional on positive purchase frequencies. An econometric model is developed to estimate the two systems by a Bayesian estimation method, which allows for an unrestricted covariance structure within each system. An empirical example
demonstrates how the estimated model can be used to formulate a profitable loss-leader pricing strategy for fish in France.
Keywords: Demand systems, fish, purchase frequencies, multivariate gamma and truncated Poisson log-normal distributions.
JEL: C34, D12, M31.
1. INTRODUCTION
A widely-used marketing strategy is the loss-leader pricing strategy (e.g., Kemp, 1955;
Hess and Gerstner, 1987; Lal and Matutes, 1994; Ellison, 2005; Chen and Rey, 2012; In and Wright, 2014). This strategy implies that a retailer reduces the price of a product below its marginal cost to attract customers to the store. In many situations, consumers buy multiple categories of goods and find it convenient to buy them from a single store.
Therefore, a decrease in the price of one category may lead a customer to transfer many of their category purchases to this store (Thomassen et al., 2017). According to Chen and Rey (2012), the loss-leader strategy is mainly used by large retailers who are
3
competing with smaller retailers with a limited variety of products. In this situation, large retailers may find it profitable to sell some products for prices below the marginal costs, and other products for prices above the marginal costs.1
Knowledge about purchase frequencies and purchased quantities can be used to formulate profitable loss-leader pricing strategies. For example, by reducing the price of a frequently-purchased product that is price elastic in the frequency of purchase, a retailer can attract more customers to the store. By simultaneously increasing the price of another product with high and positive cross-price elasticity in the frequency of purchase with respect to the first product, the retailer can increase profits.
Meghir and Robin (1992) presented a model for a consumer who chooses the purchasing frequency as well as the purchased quantities on each purchasing occasion.
They assumed that any zero purchases are generated from infrequency of purchases and would be recorded as positive purchases given a longer observation period.
Consequently, observed purchases differ from desired purchases, so they developed a two-step estimator to estimate desired purchases. In the first step, frequencies of purchase were estimated by a generalized linear model to obtain weight parameters,2 which were used to calculate the desired purchases. In the second step, the desired purchases were used to estimate the purchased quantities within a demand system.
1 Consumer purchase frequencies have frequently been modeled by count data models such as the Poison
and negative binomial models to estimate brand success, brand loyalty and store choice (e.g., Keng and Ehrenberg, 1984; Uncles et al., 1995; Bhattacharya, 1997; Uncles and Lee, 2006). Count data models such as the Poisson and negative binomial models have also been used to study demand issues in environmental economics health economics, and finance. Examples include Smith (1988), Creel and Loomis (1990) and Hellerstein (1991), who all used count data models in the estimation of the demand for recreation; Deb and Trivedi (2002), who estimated the number of doctor visits; Munkin and Trivedi (1999) and Wang (2003), who estimated the demand for health care; and Davutyan (1989), who estimated the elasticities of important factors such as bank failures. However, a demand system was not used in any of these applications.
2 Generally, a Poisson or negative binomial model would be preferred for such an estimation.
4
We extend the work of Meghir and Robin (1992) in two ways. First, they did not derive their model from a constrained optimization problem, so we develop a theoretical model based on a constrained optimization problem. Our model results in a frequency of purchase model of the form that was used by Meghir and Robin (1992). We follow them and divide the consumer’s purchase decision into a decision of frequency of purchase and a decision of quantity to purchase conditional on the purchase frequency.
Total purchased quantity of a good is then given as the product of the frequency of purchase and the average quantity purchased. The consumer’s choice variables are therefore how often to buy different products and how much to buy, on average, of the products on each occasion. In our model, homogeneity of degree zero holds for
purchase frequencies, average quantities and total quantities. Furthermore, the associated matrices of substitution effects are symmetric.
Second, Meghir and Robin (1992) focused on how to adjust for the frequency of purchase to obtain consistent parameter estimates in a demand system under an
infrequency of purchase assumption.3 Their frequency of purchase model was a basic Poisson system. In this system, homogeneity and symmetry were not imposed, the problem of zero inflation was not addressed4 and the covariance structure was assumed to be zero. We extend their count data estimation framework by: (i) accounting for homogeneity and symmetry in the frequency of the purchase demand system; (ii)
3 In applied demand analysis for disaggregative products, zero purchases represent a major problem.
Usually, it is assumed that zero purchases either represent traditional corner solutions, infrequency of purchase or non-preference for a product (e.g., Wales and Woodland, 1983; Tiffin and Arnoult, 2010). To account for zero purchases in demand system estimation, a two-step model is usually estimated. In the first step, it is estimated whether a product is purchased. The results of this step are used to correct the estimated demand system in the second step to obtain consistent parameter estimates. However, the actual frequencies of purchase are not estimated.
4 When the share of zeros in a data set significantly exceeds the amount predicted by conventional distributions, such as the Poisson, it is called zero inflation.
5
accounting for zero inflation by assuming a truncated data-generating process for the counts; and (iii) allowing for an unrestricted covariance structure within the two demand systems. We assume a truncated multivariate Poisson log-normal (TMPLN) distribution for the counts and a multivariate gamma log-normal (MGLN) distribution for the average quantities. To estimate these distributions, we use Bayesian estimation methods, specifically a random walk Metropolitan simulation algorithm.
We provide an empirical illustration on the potential usefulness of this type of model. Our example uses French scanner data for purchases of fresh fish, and the demand system includes fresh salmon, fresh white fish and other fresh fish. Our results indicate that fresh salmon is a good loss-leader candidate. Large retailers who sell a wide variety of products, including a large selection of fresh white fish, could price salmon below its marginal cost and advertise it to attract customers to the store, and then discreetly price other products above the market price to compensate for the loss on salmon. Furthermore, a smaller retailer who sells only fresh salmon and white fish may be unable to compete with these prices and subsequently driven out of the market.
This paper is organized as follows. In Sections 2 and 3, our theoretical and statistical models are developed, respectively. In Section 4, our data set and empirical specifications are described. In Section 5, our empirical results are presented and an illustrative example on how the estimated values can be used to formulate a profitable loss-leader pricing strategy is provided. In Section 6, we conclude.
2. THEORETICAL MODEL
We follow the specification and notation in Meghir and Robin (1992) as far as possible.
However, we divide the decision of how much to purchase of each good 𝑥 =
6
(𝑥1, 𝑥2, … , 𝑥𝑀) into two parts: the frequencies of purchase 𝑛 = (𝑛1, 𝑛2, … , 𝑛𝑀) and the average quantities purchased on each occasion, 𝑞 = (𝑞1, 𝑞2, … , 𝑞𝑀). By definition, the identity 𝑥(𝑛𝑞) ≡ 𝑛𝑖𝑞𝑖 holds for 𝑖 = 1,2, … , 𝑀.5 The consumer is assumed to obtain utility from the purchased goods, leisure l and frequency of purchase, and the utility function is specified as 𝑣(𝑥, 𝑛, 𝑙) = 𝑣(𝑛𝑞, 𝑛, 𝑙) where 𝑛𝑞 = (𝑛1𝑞1, 𝑛2𝑞2, … , 𝑛𝑀𝑞𝑀). The utility function is assumed to be strictly quasiconcave in n, q and l.6
The consumer has wage income 𝑦 = 𝑤𝑘, where w is the hourly wage rate and k is the number of hours spent at work. The consumer may also have other types of income R, which we assume is exogenously given. The consumer’s budget constraint is 𝑦 + 𝑅 = 𝑝′𝑛𝑞, where 𝑝 = (𝑝1, 𝑝2, … , 𝑝𝑀) is the price vector. The consumer has a constant time endowment T, which can be allocated between leisure, work and purchasing goods. The time spent on purchasing goods is given by the function 𝑔(𝑛), which is assumed to be increasing in n.7The consumer’s time constraint is 𝑇 = 𝑙 + 𝑘 + 𝑔(𝑛). The consumer’s utility maximization problem is specified as:
max𝑛,𝑞,𝑙{𝑣(𝑥(𝑛𝑞), 𝑛, 𝑙): 𝑤𝑇 + 𝑅 = 𝑝′𝑛𝑞 + 𝑤𝑙 + 𝑤𝑔(𝑛), 𝑛 > 0, 𝑞 > 0, 𝑙 > 0}. (1) It follows from the first-order conditions of Equation (1) that the solution satisfies:8
5 We assume that n is a latent variable of actual observed purchase frequencies, which can take any positive value, and, consequently, we can take the derivatives with respect to n.
6 Meghir and Robin (1992) provide two arguments for positive marginal utilities in frequencies of purchase. First, there is a benefit in the form of saved space by having to hold smaller stocks of various goods. Second, freshness is important for many types of food, including fish. The costs of frequent purchases will be reflected as lost leisure time.
7 The consumer may purchase many goods at each occasion, and the function 𝑔(𝑛) is increasing at a decreasing rate with the number of products purchased.
8 For simplicity, corner solutions are not considered in the theoretical model. The zeros in the data are assumed to be generated by infrequency of purchase and non-preference. The derivation and
interpretation of the first-order conditions of Equation (1) are given in Appendix 1. Furthermore, the wage rate is assumed to be equal to the opportunity cost of time, which is spent on either leisure or shopping.
7
𝜕𝑣 𝜕𝑥⁄ 𝑖
𝜕𝑣 𝜕𝑥⁄ 𝑗= 𝑝𝑖
𝑝𝑗 ∀𝑖, 𝑗 (2)
𝜕𝑣/𝜕𝑙
𝜕𝑣/𝜕𝑞𝑖 = 𝑤
𝑝𝑖 ∀𝑖, 𝑗 (3)
𝜕𝑣 𝜕𝑛⁄ 𝑖
𝜕𝑣 𝜕𝑙⁄ = 𝜕𝑔
𝜕𝑛𝑖 ∀𝑖 (4)
𝜕𝑣 𝜕𝑛⁄ 𝑖
𝜕𝑣 𝜕𝑛⁄ 𝑗 =𝜕𝑔 𝜕𝑛⁄ 𝑖
𝜕𝑔 𝜕𝑛⁄ 𝑗 ∀𝑖, 𝑗. (5)
Equations (2) and (3) are the standard first-order conditions of the consumer utility maximization problem with leisure. Equation (4) implies that the ratio between the marginal utility of shopping frequency for good i and the marginal utility of leisure equals the marginal cost of time spent on purchasing good i. Equation (5) implies that the ratio of marginal utilities of shopping frequencies for goods i and j are equal to the ratio of marginal costs of time spent on purchasing goods i and j.
The solution of the first-order conditions of Equation (1) results in three sets of uncompensated demand functions: 𝑛(𝑝, 𝑤, 𝑅), 𝑞(𝑝, 𝑤, 𝑅) and 𝑙(𝑝, 𝑤, 𝑅). The total purchased quantities, 𝑥(𝑝, 𝑤, 𝑅), are found by substituting 𝑛(𝑝, 𝑤, 𝑅) and 𝑞(𝑝, 𝑤, 𝑅) into the identity 𝑥(𝑛𝑞) ≡ 𝑛𝑖𝑞𝑖.
The dual problem to the consumer’s utility maximization problem (1) is:
min𝑛,𝑞,𝑙{𝑅 = 𝑝′𝑛𝑞 − 𝑤(𝑇 − 𝑙 − 𝑔(𝑛)): 𝑣(𝑥(𝑛𝑞), 𝑛, 𝑙) = 𝑣∗, 𝑛 > 0, 𝑞 > 0, 𝑙 > 0}. The solution to the first-order conditions of this problem results in three sets of compensated demand functions: 𝑛𝑐(𝑝, 𝑤, 𝑣∗), 𝑞𝑐(𝑝, 𝑤, 𝑣∗) and 𝑙𝑐(𝑝, 𝑤, 𝑣∗).9 The compensated
9 The derivation of the first-order conditions of the minimization problem is given in Appendix 1.
8
demand functions for total purchased quantities, 𝑥𝑐(𝑝, 𝑤, 𝑣∗), are found by substituting 𝑛𝑐(𝑝, 𝑤, 𝑣∗) and 𝑞𝑐(𝑝, 𝑤, 𝑣∗) into the identity 𝑥𝑖𝑐 ≡ 𝑛𝑖𝑐𝑞𝑖𝑐.
The conditions for homogeneity of degree zero in prices and income, symmetry and negativity for the compensated and uncompensated demand functions are
summarized in:10 Proposition 1.1
The demand equations 𝑛(𝑝, 𝑤, 𝑅), 𝑞(𝑝, 𝑤, 𝑅), 𝑙(𝑝, 𝑤, 𝑅) and 𝑥(𝑝, 𝑤, 𝑅) are homogeneous of degree zero in (𝑝, 𝑤, 𝑅).
Proposition 1.2
The matrix of compensated substitution effects for 𝑥𝑐(𝑝, 𝑤, 𝑣∗) is symmetric and negative semidefinite.
Proposition 1.3
The matrix of compensated substitution effects for the product 𝑛𝑐(𝑝, 𝑤, 𝑣∗)𝑞𝑐(𝑝, 𝑤, 𝑣∗) is symmetric and negative semidefinite.
3. STATISTICAL MODEL
The frequency of shopping 𝑛𝑖 = (𝑛𝑖11, 𝑛𝑖12, … , 𝑛𝑖𝐾𝑇) is assumed to follow a discrete distribution, 𝑓𝑁𝑖(𝑛𝑖|𝛽𝑖, 𝐶), for 𝑛 = 0,1,2, … where 𝛽𝑖 is a vector of parameters and C is a matrix of explanatory variables. Other basic notations are as follows: N represents a random variable and 𝑛𝑖𝑘𝑡 is an observed value of N, where the subscript i denotes the product; 𝑘 = 1,2, … , 𝐾 denotes the household; and 𝑡 = 1,2, … , 𝑇 denotes the time period. The average purchases 𝑞𝑖 = (𝑞𝑖11, 𝑞𝑖12, … , 𝑞𝑖𝐾𝑇) are only observed after a trip to
10 These propositions are proven in Appendix 1.
9
the shop. Thus, the variable 𝑞𝑖|𝑛𝑖 > 0 is assumed to follow a continuous distribution, 𝑓𝑄𝑖|𝑛𝑖>0(𝑞𝑖|𝛼𝑖, 𝐶, 𝑛𝑖 > 0), defined only over positive values, where 𝛼𝑖 is a vector of parameters. The interpretations of Q and 𝑞𝑖𝑘𝑡 are as for N and 𝑛𝑖𝑘𝑡. The data-generating process for the average quantity purchased is represented by the following two-part model:
𝑓𝑄(𝑞𝑖|𝛼𝑖, 𝐶) = ( Pr(𝑁 = 0|𝛽𝑖, 𝐶) if 𝑞𝑖 = 0
Pr(𝑁 > 0|𝛽𝑖, 𝐶)𝑓𝑄𝑖|𝑛>0(𝑞𝑖|𝛼𝑖, 𝐶, 𝑛𝑖 > 0) if 𝑞𝑖 > 0) (6) The decisions to purchase a good and how much to purchase in each trip are likely related, and it is desirable to model them as stochastically correlated. Furthermore, the demand for one good may be related to the demand for other goods, and it is important to allow for correlation between equations within each of the two systems. To allow for these correlations, random effects are introduced to both densities, 𝑓𝑁𝑖(𝑛𝑖|𝛽𝑖, 𝐶, 𝑏𝑁𝑖) and 𝑓𝑄𝑖|𝑛>0(𝑞𝑖|𝛼𝑖, 𝐶, 𝑛 > 0, 𝑏𝑄𝑖), where 𝑏𝑁𝑖 and 𝑏𝑄𝑖 are random effects assumed to follow a multivariate normal distribution:
[𝑏𝑁
𝑏𝑄]| 𝐷 ∼ 𝑀𝑉𝑁 ([0𝑀
0𝑀] , [𝐷𝑁 𝐷𝑁𝑄
𝐷𝑁𝑄 𝐷𝑄]) (7)
where 𝑏𝑁 = (𝑏𝑁1, … , 𝑏𝑁𝑀), 𝑏𝑄 = (𝑏𝑄1, … , 𝑏𝑄𝑀) and D is an unrestricted block- covariance matrix.11 The joint probability density function for ni and qi is given as:
𝑝(𝑛𝑖, 𝑞𝑖|𝛽, 𝛼, 𝐷, 𝐶) =
11 The conditional means of 𝑛𝑖 and 𝑞𝑖 are given as follows: E(𝑛𝑖|𝛽𝑖, 𝐶, 𝑏𝑁𝑖) and E(𝑞𝑖|𝛼𝑖, 𝐶, 𝑏𝑄𝑖) = Pr(𝑁 > 0|𝛽𝑖, 𝐶)E(𝑞𝑖|𝛼𝑖, 𝐶, 𝑛𝑖> 0, 𝑏𝑄𝑖). The marginal effects of E(𝑞𝑖|𝛼𝑖, 𝐶, 𝑏𝑄𝑖) are given by:
𝜕E(𝑞𝑖|𝛼𝑖, 𝐶, 𝑏𝑄𝑖)
𝜕𝐶𝑖 =𝜕E(𝑞𝑖|𝛼𝑖, 𝐶, 𝑛𝑖> 0, 𝑏𝑄𝑖)
𝜕𝐶𝑖 Pr(𝑁 > 0|𝛽𝑖, 𝐶). The corresponding elasticities are given by:
𝜕E(𝑞𝑖|𝛼𝑖, 𝐶, 𝑏𝑄𝑖)
𝜕𝐶𝑖
𝐶𝑖
𝑞𝑖, and 𝜕E(𝑛𝑖|𝛽𝑖, 𝐶, 𝑏𝑁𝑖)
𝜕𝐶𝑖
𝐶𝑖 𝑛𝑖.
10
∫ ∏ 𝑓𝑁𝑖(𝑛𝑖𝑘𝑡|𝛽𝑖, 𝐶, 𝑏𝑁𝑖𝑘𝑡)
𝑇
𝑡=1
𝑓𝑄𝑖|𝑛>0(𝑞𝑖|𝛼𝑖, 𝐶, 𝑛𝑖 > 0)𝜙(𝑏𝑖𝑘|0, 𝐷)𝑑𝑏𝑖𝑘.
(8)
The product operator is inside the integral because 𝑏𝑁 and 𝑏𝑄 each have one draw for the 2T random variables 𝑛𝑖𝑘1, 𝑛𝑖𝑘2, … , 𝑛𝑖𝑘𝑇 and 𝑞𝑖𝑘1, 𝑞𝑖𝑘2, … , 𝑞𝑖𝑘𝑇, respectively. Thus, there is a new draw for each cluster, but not for each time period within a cluster. The likelihood is then given by:
𝐿 = ∏ ∏ 𝑝(𝑛𝑖, 𝑞𝑖|𝛽𝑖, 𝛼𝑖, 𝐷, 𝐶)
𝑀
𝑖=1 𝐾
𝑘=1
(9)
Because the joint density 𝑝(𝑛𝑖, 𝑞𝑖|𝛽𝑖, 𝛼𝑖, 𝐷, 𝐶) does not have a closed form solution, the likelihood L cannot be optimized with conventional Newton methods, and we therefore use simulation methods.
3.1. Distribution Assumptions
To account for the large share of zeros in the data, we assume that the frequency of purchase, 𝑛𝑖, is generated from 𝑛𝑖|𝑛𝑖 > 0, 𝛽𝑖, 𝐶, ∼ truncated Poisson(𝜇𝑖), where 𝜇𝑖 = exp(𝐶𝛽𝑖+ 𝑏𝑁𝑖). This results in a multivariate truncated Poisson log-normal
distribution:
𝑓𝑁𝑖(𝑛𝑖|𝑛𝑖 > 0, 𝛽, 𝐷𝑁, 𝐶) = ∫exp(−𝜇𝑖(𝑏𝑁𝑖))(𝜇𝑖(𝑏𝑁𝑖))𝑛𝑖
[1 − exp(−𝜇𝑖(𝑏𝑁𝑖))]𝑛𝑖! 𝜙(𝑏𝑁𝑖|0, 𝐷𝑁)𝑑𝑏𝑁𝑖 (10) where 𝜙(𝑏𝑁𝑖|0, 𝐷𝑁) is the multivariate normal distribution for 𝑏𝑁𝑖, with covariance matrix 𝐷𝑁. We assume that the average quantity of purchase, 𝑞𝑖, is generated from 𝑞𝑖|𝛼𝑖, 𝐶, 𝑛𝑖 > 0 ∼ Gamma(𝜅𝑖, 𝜂𝑖), where the mean of the gamma distribution is
specified as 𝜅𝑖𝜂𝑖 = exp(𝐶𝑖𝛼𝑖 + 𝑏𝑄𝑖). This specification results in a multivariate gamma log-normal distribution:
11
𝑓𝑄𝑖(𝑞𝑖|𝛼𝑖, 𝐶, 𝑛𝑖 > 0) = ∫𝑞𝑖𝜅𝑖−1exp(−𝑞𝑖⁄ )𝜂𝑖
𝜂𝑖𝜅Γ(𝜅𝑖) 𝜙(𝑏𝑄𝑖|0, 𝐷𝑄)𝑑𝑏𝑄𝑖 (11) where 𝜙(𝑏𝑄𝑖|0, 𝐷𝑄) is the multivariate normal distribution for 𝑏𝑄𝑖, with covariance matrix 𝐷𝑄.
3.2 Priors and Markov Chain Monte Carlo Sampling
We assume uninformative priors, which is a common practice (see, e.g., Chib and Winkelmann, 2001). Let 𝛽 ∼ N(𝛽0, 𝐵0−1), 𝛼 ∼ N(𝛼0, 𝐴0−1), 𝜅 ∼ Gamma(𝑘0, 𝑠0), 𝐷𝑁−1 ∼ Wishart (𝑣𝑁0, 𝑅𝑁0), and 𝐷𝑄−1∼ Wishart (𝑣𝑄0, 𝑅𝑄0), where
𝛽0, 𝐵0, 𝛼0, 𝐴0, 𝑘0, 𝑠0, 𝑣𝑁0, 𝑅𝑁0, 𝑣𝑄0, and 𝑅𝑄0 are known hyperparameters, and
Wishart(𝑣𝑜0, 𝑅𝑜0) is the Wishart distribution with 𝑣𝑜0 degrees of freedom and a scale matrix 𝑅𝑜0, where 𝑜 = 𝑁, 𝑄. By Bayes’ theorem, the posterior density of the two parts of the model are proportional to the following expressions:
𝜙(𝛽|𝛽0, 𝐵0−1)𝑓𝑊(𝐷𝑁−1|𝑣𝑁0, 𝑅𝑁0) ∏ ∏ 𝑓𝑁(𝑛𝑖𝑘|𝛽, 𝑏𝑁𝑖𝑘)
𝑀
𝑖=1 𝐾
𝑘=1
𝜙(𝑏𝑁𝑖𝑘|0, 𝐷𝑁)
(12)
𝜙(𝛼|𝛼0, 𝐴−10 )𝑓𝑊(𝐷𝑄−1|𝑣𝑄0, 𝑅𝑄0) ∏ ∏ 𝑓𝑄(𝑞𝑖𝑘|𝛼, 𝑏𝑄𝑖𝑘)
𝑀
𝑖=1 𝐾
𝑘=1
𝜙(𝑏𝑄𝑖𝑘|0, 𝐷𝑄)
(13)
where 𝑓𝑊 is the Wishart density. We construct Markov chains using the blocks of parameters 𝑏𝑁, 𝑏𝑄, 𝛽, 𝛼, 𝐷𝑁 and 𝐷𝑄 and the full conditional distributions:
[𝑏𝑁|𝑛, 𝛽, 𝐷]; [𝛽|𝑛, 𝑏𝑁]; [𝐷|𝑏𝑁] (14) [𝑏𝑄|𝑞, 𝛼, 𝐷]; [𝛼|𝑞, 𝑏𝑄]; [𝐷|𝑏𝑄] (15) The blocks of parameters [𝛽|𝑛, 𝑏𝑁] and [𝛼|𝑞, 𝑏𝑄] are separated into smaller blocks to facilitate convergence. The simulation output is generated by recursively simulating
12
these distributions using the most recent values of the conditioning variables at each step. The mathematical derivation of the Markov chain Monte Carlo sampling of 𝑏𝑁, 𝑏𝑄, 𝛽, 𝛼, 𝐷𝑁 and 𝐷𝑄 can be found in Appendix 2.
4. DATA AND SELECTED FUNCTIONAL FORM
French scanner data for the purchases of fresh fish as recorded by Kantar Worldpanel for the period 2005–2008 are used. The data were from a rotating panel, i.e., when households drop out, new households are selected to take their place. The numbers of participants recording fresh fish purchases were 3,291 in 2005, 3,234 in 2006, 3,165 in 2007 and 4,479 in 2008, which show significant rotation of households in 2008.
Purchased quantities and total expenditures were recorded. To calculate the associated unit prices, we followed Allais et al. (2010) and Bertail and Caillavet (2008) and
divided the total expenditures by quantities. The data are weekly and include many zero purchases. To reduce the number of zero purchases, weekly purchases were aggregated to yearly purchases, and the panel structure was accounted for by using random effects, as discussed in the previous section. Demand for salmon has been increasing during recent years and along with white fish, if canned tuna is excluded, has been the most frequently purchased type of fish in France, while fresh fish has been the most common product form (Xie and Myrland, 2011). Therefore, we included fresh salmon, fresh white fish and other fresh fish in our demand system.
In Table I, the purchase frequencies for fresh salmon, fresh white fish and other fresh fish over the 4-year survey period are presented. Between 14% and 45% did not purchase each of the three types of fish. Many households with a positive purchase of one fish type only purchased this type once or twice over the 4 years. However, more
13
than 10% of the households purchased white fish more than 10 times and almost 24%
purchased other fish more than 10 times.
(Table I about here)
Table II shows summary statistics of the variables used in our empirical model.
The average frequencies of purchase and the average quantities were quite low because of the large share of zeros in the data set. However, the maximum frequency was 172 times for the other fish. The average purchase of each type of fish was around 500 g and the maximum average purchase of each type of fish was around 9 kg. The unit prices and total expenditures on fresh fish were divided by the average French consumer price index (CPI) to impose homogeneity of degree zero for frequencies of purchases,
average quantities and total quantities.12
(Table II about here) Total quantities purchased are specified as:
𝑥𝑖 = exp(𝑏(𝑝, 𝑤, 𝑅, 𝜃)) (16)
where the function 𝑏(⋅) is linear in its parameters 𝜃 = (𝜃1, 𝜃2, … , 𝜃𝑛). We decomposed 𝑏(⋅) into a frequency part 𝑓(𝑝, 𝑤, 𝑅, 𝛽) and an average quantity part 𝑧(𝑝, 𝑤, 𝑅, 𝛼), where 𝛽 = (𝛽1, 𝛽2, … , 𝛽𝑛) and 𝛼 = (𝛼1, 𝛼2, … , 𝛼𝑛) are parameter vectors. Using this decomposition, Equation (16) becomes:
𝑥𝑖 = 𝑛𝑖𝑞𝑖 = exp(𝑓(𝑝, 𝑤, 𝑅, 𝛽))exp(𝑧(𝑝, 𝑤, 𝑅, 𝛼)). (17)
12 The average CPI for 2005–2008 was 90.28, and the relative price of 0.13 for salmon and 0.14 for white
fish reported in Table 2 correspond to €11.74 and €12.64 per kg of salmon and white fish, respectively.
14
As is conventional when estimating count data models, the conditional
expectation is defined as a semi-logarithmic function. The purchase frequency demand function for each fish type is thus given by:
E(𝑛𝑖𝑘|𝐶𝑖) = exp (𝛽𝑖𝑘+ ∑ 𝛽𝑖𝑠(𝑝𝑠𝑘/𝐶𝑃𝐼)
𝑀
𝑠=1
+ 𝜉𝑖((𝑅𝑘+ 𝑦𝑘)/𝐶𝑃𝐼) + 𝑏𝑁𝑖𝑘)
(18)
for 𝑖 = 1,2, … , 𝑀 fish categories and 𝑘 = 1,2, … , 𝐾 households. The price of fish category s for household k is denoted by 𝑝𝑠𝑘. The total fish expenditure of household k is given by 𝑅𝑘+ 𝑦𝑘, where CPI is the French consumer price index, and 𝑏𝑁𝑖𝑘 is the random effect described in Section 3. For the demand system to be consistent with economic theory, homogeneity and symmetry were imposed on the parameters.13 LaFrance and Hanemann (1989) and LaFrance (1990) derived these restrictions for the semi-logarithmic functional form. In our case, we must have 𝛽𝑖𝑠 = 0 ∀ 𝑖 ≠ 𝑠 and 𝜉𝑖 = 𝜉 ∀ 𝑖. To allow for additional explanatory variables 𝑐𝑧𝑘, we define 𝛽𝑖𝑘 as follows:
𝛽𝑖𝑘 = 𝜂0𝑖+ ∑ 𝑐𝑧𝑘𝜂𝑖𝑧
𝑍
𝑧=1
(19)
The average quantity demand function for each good is given by:
E(𝑞𝑖𝑘|𝐶𝑖) = exp (𝛼𝑖𝑘+ ∑ 𝛼𝑖𝑠(𝑝𝑠𝑘/𝐶𝑃𝐼)
𝑀
𝑠=1
+ 𝛾𝑖((𝑅𝑘+ 𝑦𝑘)/𝐶𝑃𝐼) + 𝑏𝑄𝑖𝑘).
(20)
Corresponding to the restrictions on Equation (18), the restrictions 𝛼𝑖𝑠 = 0 ⩝ 𝑖 ≠ 𝑠 and 𝛾𝑖 = 𝛾 ∀ 𝑖 were imposed on Equation (20), and 𝛼𝑖𝑘 is defined as 𝛽𝑖𝑘 in Equation (19).
13 Negativity was not imposed due to computational difficulties in that it required non-negativity
constraints.
15
The own-price 𝑒𝑛𝑖 and total expenditure elasticities 𝜀𝑛𝑖 for the frequency of purchase were calculated as:
𝑒𝑛𝑖 =𝜕E(𝑛𝑖|𝐶𝑖)
𝜕𝑝𝑖 𝑝𝑖
𝑛𝑖 = 𝛽𝑖𝑖𝑝𝑖 (21)
𝜀𝑛𝑖 = 𝜕E(𝑛𝑖|𝐶𝑖)
𝜕(𝑅 + 𝑦)
(𝑅 + 𝑦)
𝑛𝑖 = 𝜃(𝑅 + 𝑦). (22)
The own-price 𝑒𝑞𝑖 and total expenditure 𝜀𝑞𝑖 elasticities for the average quantity purchased were calculated as:
𝑒𝑞𝑖 =𝜕E(𝑞𝑖|𝐶𝑖)
𝜕𝑝𝑖 𝑝𝑖
𝑛𝑖 = 𝛼𝑖𝑖𝑝𝑖 (23)
𝜀𝑞𝑖 =𝜕E(𝑞𝑖|𝐶𝑖)
𝜕(𝑅 + 𝑦)
(𝑅 + 𝑦)
𝑛𝑖 = 𝛾(𝑅 + 𝑦). (24)
Under the restrictions above, all the total substitution effects are zero, and the uncompensated cross-price elasticities are also zero.14 However, the compensated price elasticities are not zero. These elasticities can be useful for calculating welfare effects, and the elasticity formulas and the estimated values are provided in Appendix 1. In our case, all of the compensated cross-price elasticities were very small numerical values.
5. EMPIRICAL RESULTS
Table III shows the posterior summary for the fresh salmon equations based on the TMPLN and MGLN distributions. The purchase frequencies were estimated by the TMPLN model, and the average purchases were estimated by the MGLN model. Only
14 No total substitution effects follow from the imposition of symmetry on a semi-logarithmic functional
form. It might be of interest for future research to estimate a system without symmetry restrictions to compare the differences between the compensated and uncompensated cross-price elasticities.
16
the signs and statistical significances of the estimated parameters in the table have an interpretation.
All parameter estimates in the frequency of purchase model were significant except for the time dummies for 2007 and 2008. The signs of the estimated parameters were as expected. The time dummy for 2006 shows that the frequency of purchase was smaller than that in 2005. The Geweke Z-test of Markov chain stationarity was not rejected at the 5% level except for the time dummy for 2006.15
All of the parameter estimates in the equation for average purchases were significant and were generated from a stationary Markov process. The time dummies show that average quantities have increased for all years, relative to 2005. Kappa is the shape parameter of the gamma distribution and was highly significant.16
Table IV shows the posterior summary for the fresh white fish equations. All parameters in the frequency of purchase model were significant except for the 2008 dummy. Furthermore, we did not reject that they were generated from a stationary Markov process. The time dummies showed that the frequency of purchase decreased relative to 2005. All parameter estimates in the equation for average purchases were significant and their Markov chains were stationary, except for the constant. As for salmon, the time dummies showed an increase relative to 2005.
Table V shows the posterior summary for the other fresh fish equations. All of the parameter estimates of both models were significant and were generated from a
15 The Geweke Z-values are normally distributed. For an application of the Geweke convergence test, see, e.g., Nylander et al. (2008).
16 Many probability distributions are not a single distribution, but a family of distributions with one or
several shape parameters, which allow the distributions to take on different shapes. These distributions are very useful in modeling since they are flexible and can fit many different datasets.
17
stationary Markov process. The frequency of purchase decreased over time, whereas the average quantity demanded increased relative to 2005.
Table VI shows the cross-equations covariance matrix for the models, i.e., the covariance matrix of the random effects. We define the demand equation for salmon as number 1, the demand for white fish as number 2 and the demand for other fish as number 3. Then, Sigma11 denotes the variance of the random effects for salmon, Sigma22 for white fish and Sigma33 for other fish. Correspondingly, Sigma12 denotes the covariance between the random effects of equations one and two and so on.
All parameter estimates in both systems were significant and were generated from a stationary Markov process. The significant covariance estimates demonstrated the importance of allowing for an unrestricted covariance matrix.
Table VII shows the own-price and total expenditure elasticities with respect to frequencies of purchase, average purchases and total purchases. The values of the own- price elasticities for total quantities purchased are almost equal for the three types of fish, whereas the own-price elasticities for purchase frequencies and average quantities are more unequal. Such differences could be used to find a loss-leader product.
(Tables III–VII about here)
Example of a Loss-Leader Pricing Strategy
We include a simple example showing how fresh salmon could be used as a loss-leader product. We assume a retailer has a large variety of goods and that they want to create a profitable pricing strategy that also benefits their competitive position relative to nearby fish retailers. Let the retailer lower the price of salmon by 20% and raise the price of fresh white fish by 20%. As described above, the uncompensated cross-price effects
18
would be zero in our estimated system, and we only have to take account of the own- price effects. However, as discussed in the meta-study of Cornelsen et al. (2016), the cross-price effects are likely to be much smaller than the own-price effects for many food products, and they will only reinforce or work in the opposite direction of the own- price effects.
The key numbers and calculations used in the example are summarized in Table VIII. As reported in Table VII, the own-price elasticity with respect to frequency of purchase for salmon was –0.05, and a 20% price reduction would increase the frequency of purchase by 1% from 1.96 to 1.98. The own-price elasticity with respect to average quantity was –0.18, and a 20% price reduction would increase the average quantity purchased from 421.78 to 436.96 g. For white fish, the 20% price increase would reduce the frequency of purchase from 3.77 to 3.69 and reduce the average quantity from 435.03 to 425.46 g.
As shown in Table II, the sample averages of the relative prices of white fish and salmon were 0.14 and 0.13, respectively. Multiplying these relative prices by the
average CPI of 90.28 resulted in nominal average prices of €12.64 for white fish and
€11.74 for salmon. After the 20% price reduction for salmon, the new price became
€9.39, and after the 20% price increase for white fish, the new price became €15.17.
The per customer revenue of sales of salmon given the initial price was €9.71, while it was €8.12 after the 20% reduction in price. The per customer revenue of sales of white fish given the initial price was €20.73, while it was €23.82 after the 20%
increase in price. The per customer total revenue from sales of salmon and white fish increased from €30.44 to €31.94, i.e., an increase of 4.9%. Given that no changes in
19
total costs were caused by these price changes, the profit increased by €1.50 per customer.
The above effect on revenue only included the direct effects on revenue from sales of the two categories of fish. However, this loss-leader pricing strategy may also convince customers to transfer more of their other category purchases to this store (Thomassen et al., 2017). Most customers not only purchase fresh fish, but also may increase purchases of other products such as meat, bakery, fruits, or toothpaste.
(Table VIII about here)
6. CONCLUSIONS
The microeconomic model presented by Meghir and Robin (1992) assumed that total purchased quantities could be expressed as the product of the average purchased quantities and the frequencies of purchase. We follow their specification and further develop the model so that homogeneity of degree zero and symmetry hold for purchase frequencies, average purchased quantities and total purchased quantities. Furthermore, we extend their count data estimation framework in three ways: (i) we account for homogeneity and symmetry in the count data demand system; (ii) we account for the problem of zero purchases by assuming a truncated data-generating process for the counts; and (iii) we allow the covariance structure to be unrestricted within the two demand systems. We assume a multivariate Poisson log-normal distribution for the counts and a multivariate gamma log-normal distribution for the average quantities. To estimate these distributions, we use a random-walk Metropolitan simulation algorithm.
The proposed estimation method is illustrated by using scanner data for French fish purchases to estimate the demand for fresh salmon, fresh white fish and other fresh
20
fish. The results indicate that salmon can be a loss-leader category for large retailers who want to increase store traffic and, thereby, sales of products that can be sold with a markup above the market price. Future research will focus on providing a more
elaborate empirical example of the loss-leader pricing strategy with different product groups.
REFERENCES
Allais, O., Bertail, P., & Nichele V. (2010). The effect of a fat tax on French
households’ purchases: a nutrition approach. American Journal of Agricultural Economics, 92, 228–245.
Bertail, P., & Caillavet, F. (2008). Fruit and vegetable consumption patterns: a
segmentation approach. American Journal of Agricultural Economics, 90, 827–
842.
Bhattacharya, C.B. (1997). Is your brand’ s loyalty too much, too little, or just right?
Explaining deviations in loyalty from the Dirichlet norm. International Journal of Research in Marketing, 14, 421–435.
Chen, Z., and Rey, P. (2012). Loss Leading as an Exploitative Practice. The American Economic Review 102, 3462-3482.
Chib, S., and Winkelmann, R. (2001). Markov Chain Monte Carlo Analysis of
Correlated Count Data. Journal of Business & Economic Statistics 19, 428-435.
Creel, M.D., and Loomis, J.B. (1990). Theoretical and Empirical Advantages of Truncated Count Data Estimators for Analysis of Deer Hunting in California.
American Journal of Agricultural Economics 72, 434–441.
21
Cornelsen, L., Mazzocchi, M., Green, R., Dangour, A.D., and Smith, R.D. (2016).
Estimating Relationship between Food Prices and Food Consumption – Methods Matter. Applied Economic Perspectives and Policy 38, 546-561.
Davutyan, N. (1989). Bank Failures as Poisson Variates. Economics Letters 29, 333- 338.
Deb, P., and Trivedi, P.K. (2002). The Structure of Demand for Health Care: Latent Class versus Two-Part Models. Journal of Health Economics 21, 601-625.
Ellison, G. (2005). A Model of Add-On Pricing. The Quarterly Journal of Economics 120, 585-637.
Hellerstein, D.M. (1991). Using Count Data Models in Travel Cost Analysis with Aggregate Data. American Journal of Agricultural Economics 73, 860–866.
Hess, J.D., and Gerstner, E. (1987). Loss Leader Pricing and Rain Check Policy.
Marketing Science 6, 358-374.
In, Y., and Wright, J. (2014). Loss-Leader Pricing and Upgrades. Economics Letters 122, 19-22.
Kemp, M.C. (1955). An Appraisal of Loss Leader Selling. The Canadian Journal of Economics and Political Science 21, 245-250.
Keng, K.A.K, and Ehrenberg, A.S.C. (1984). Patterns of Store Choice. Journal of Marketing Research 21, 399-409.
LaFrance, J.T., and Hanemann, W.M. (1989). The Dual Structure of Incomplete Demand Systems. American Journal of Agricultural Economics 71, 262-274.
LaFrance, J.T. (1990). Incomplete Demand Systems and Semilogarithmic Demand Models. Australian Journal of Agricultural Economics 34, 118-131.
22
Lal, R., and Matutes, C. (1994). Retail Pricing and Advertising Strategies. Journal of Business 67, 345-370.
Meghir, C., and Robin, J.M. (1992). Frequency of Purchase and the Estimation of Demand Systems. Journal of Econometrics 53, 53–85.
Munkin, M.K., and Trivedi, P.K. (1999). Simulated Maximum Likelihood Estimation of Multivariate Mixed-Poisson Regression Models, with Application. Econometrics Journal 2, 29-48.
Nylander, J.A.A., Wilgenbusch, J.C., Warren, D.L., and Swofford, D.L. (2008). AWTY (Are We There Yet?): A System for Graphical Exploration of MCMC
Convergence in Bayesian Phylogenetics. Bioinformatics 24, 581-583.
Smith, V.K. (1988). Selection and Recreation Demand. American Journal of Agricultural Economics 70, 29-36.
Tiffin, R., and Arnoult, M. (2010). The Demand for a Healthy Diet: Estimating the Almost Ideal Demand System with Infrequency of Purchase. European Review of Agricultural Economics 37, 501-521.
Thomassen, Ø., Smith, H., Seiler, S., and Schiraldi, P. (2017). Multi-Category Competition and Market Power: A Model of Supermarket Pricing. American Economic Review (Forthcoming).
Uncles, M., Ehrenberg, A.S.C., and Hammond, K. (1995). Patterns of Buyer Behavior:
Regularities, Models, and Extensions. Marketing Science 14, 71-78.
Uncles, M., and Lee, D. (2006). Brand Purchasing by Older Consumers: An
Investigation Using the Juster Scale and the Dirichlet Model. Marketing Letters 17, 17-29.
23
Wales, T.J., and Woodland, A.D. (1983). Estimation of Consumer Demand Systems with Binding Non-Negativity Constraints. Journal of Econometrics 21, 263-285.
Wang, P. (2003). A Bivariate Zero-Inflated Negative Binomial Regression Model for Count Data with Excess Zeros. Economics Letters 78, 373-378.
Xie, J., and Myrland, Ø. (2011). Consistent Aggregation in Fish Demand: A Study of French Salmon Demand. Marine Resource Economics 26, 267-280.
24 Table I. Purchase frequencies
No. of Purchases
Salmon White Fish Other Fish Frequency Percent Frequency Percent Frequency Percent
0 6326 44.64 4126 29.11 1982 13.99
1 2784 19.64 2738 19.32 2357 16.63
2 1514 10.68 1654 11.67 1603 11.31
3 1001 7.06 1135 8.01 1223 8.63
4 685 4.83 871 6.15 953 6.72
5 473 3.34 635 4.48 741 5.23
6 337 2.38 489 3.45 580 4.09
7 254 1.79 402 2.84 549 3.87
8 151 1.07 342 2.41 446 3.15
9 131 0.92 248 1.75 345 2.43
≥10 516 3.65 1532 10.84 3393 23.96
25 Table II. Summary statistics for fish purchases
Variable Mean Std. Dev. Min Max
Salmon (frequency) 1.96 3.39 0.00 52.00
White fish (frequency) 3.77 6.03 0.00 87.00
Other fish (frequency) 7.43 11.25 0.00 172.00
Salmon (average quantity in g) 421.78 608.24 0.00 9600.00 White fish (average quantity in g) 435.03 480.61 0.00 8537.70 Other fresh fish (average quantity in g) 552.08 467.96 0.00 8100.00
Real price of salmon 0.13 0.04 0.01 0.75
Real price of white fish 0.14 0.05 0.02 0.71
Real price of other fish 0.12 0.05 0.01 0.68
Real total expenditures on fresh fish 0.98 1.33 0.00 15.57
26
Table III. Posterior summary for salmon based on the TMPLN and MGLN distributions
Frequency Average Quantity
Variable Mean t-value Geweke Z Mean t-value Geweke Z
Constant –0.53 –16.00 1.07 6.66 369.87 –0.71
Price –0.37 –1.97 –1.17 –3.02 –28.73 0.61
Expenditure 0.09 62.41 1.75 0.07 19.59 –0.25
Time06 –0.05 –3.80 2.10 0.11 9.30 –0.50
Time07∙10 –0.07 –0.62 0.91 1.26 10.99 –0.03
Time08∙100 –0.06 –0.05 0.15 14.27 12.63 –0.32
Kappa - - - 2.36 136.60 –0.84
Notes: TMPLN = truncated multivariate Poisson log-normal and MGLN = multivariate gamma log- normal. Time06, Time07 and Time08 are annual dummy variables, which take the value of 1 in the indicated years. For the ease of reading, some of these dummy variables are scaled in this table. The Geweke Z provides the Z-value for a test of stationarity of the Markov chains. Kappa is the shape parameter of the gamma distribution.
27
Table IV. Posterior summary for white fish based on the TMPLN and MGLN distributions
Frequency Average Quantity
Variable Mean t-value Geweke Z Mean t-value Geweke Z
Constant –0.12 –5.94 0.35 6.35 402.04 –2.57
Price –0.69 –6.84 –0.60 –1.38 –14.40 1.70
Expenditure 0.09 62.41 1.75 0.07 19.59 –0.25
Time06 –0.04 –5.66 –0.38 0.06 5.15 0.35
Time07 –0.04 –6.18 0.57 0.05 4.56 0.75
Time08 –0.01 –1.80 –0.16 0.06 5.55 1.35
Kappa - - - 2.06 105.80 1.14
Notes: TMPLN = truncated multivariate Poisson log-normal and MGLN = multivariate gamma log- normal. Time06, Time07 and Time08 are annual dummy variables, which take the value of 1 in the indicated years. The Geweke Z provides the Z-value for a test of stationarity of the Markov chains.
28
Table V. Posterior summary for other fish based on the TMPLN and MGLN distributions
Frequency Average Quantity
Variable Mean t-value Geweke Z Mean t-value Geweke Z
Constant 0.32 24.00 –0.04 6.43 351.16 1.08
Price –0.81 –10.55 0.24 –1.22 –10.50 –1.32
Expenditure 0.09 62.41 1.75 0.07 19.59 –0.25
Time06 –0.05 –15.47 –0.41 0.06 5.06 –1.72
Time07 –0.05 –15.42 –0.80 0.11 8.99 –1.80
Time08 –0.07 –18.67 1.18 0.13 10.48 –1.69
Kappa - - - 2.22 123.39 –0.15
Notes: TMPLN = truncated multivariate Poisson log-normal and MGLN = multivariate gamma log- normal. Time06, Time07 and Time08 are annual dummy variables, which take the value of 1 in the indicated years. The Geweke Z provides the Z-value for a test of stationarity of the Markov chains.
29
Table VI. Posterior summary cross-equations covariance matrix based on TMPLN and MGLN distributions
Frequency Average Quantity
Variable Mean t-value Geweke Z Mean t-value Geweke Z
Sigma11 0.49 18.65 –0.55 0.29 37.95 0.21
Sigma12 0.17 11.53 –0.05 0.13 24.04 0.90
Sigma13 0.17 14.53 –0.92 0.31 40.83 –0.33
Sigma21 0.17 11.53 –0.05 0.13 24.04 0.90
Sigma22 0.40 23.74 –0.23 0.21 36.01 2.37
Sigma23 0.21 21.79 –0.64 0.14 23.59 0.07
Sigma31 0.17 14.53 –0.92 0.31 40.83 –0.33
Sigma32 0.21 21.79 –0.64 0.14 23.59 0.07
Sigma33 0.26 29.50 –0.90 0.34 38.48 –0.56
Note: TMPLN = truncated multivariate Poisson log-normal and MGLN = multivariate gamma log- normal. Sigma11, Sigma22 and Sigma33 represent the variances of the random effects of the demand for salmon, white fish and other fish, respectively. The other Sigmas represent the corresponding
covariances. Geweke Z provides the Z-value for a test of stationarity of the Markov chains.
30
Table VII. Own-price and total expenditure elasticities
Frequency
Salmon White Fish Other Fish Estimate t-value Estimate t-value Estimate t-value
Own price –0.05 –1.97 –0.10 –6.84 –0.10 –10.55
Total expenditure 0.09 62.41 0.09 62.41 0.09 62.41
Average Quantities
Salmon White Fish Other Fish Estimate t-value Estimate t-value Estimate t-value
Own price –0.18 –28.73 –0.11 –14.40 –0.30 –10.50
Total expenditure 0.03 19.59 0.04 19.59 0.07 19.59
Total Quantities
Salmon White Fish Other Fish Estimate t-value Estimate t-value Estimate t-value
Own price –0.24 –8.77 –0.21 –7.50 –0.21 –20.51
Total expenditure 0.12 42.65 0.13 53.85 0.14 81.57
31
Table VIII. Numerical example loss-leader pricing strategy
Frequency/Quantity Revenue Product Elasticitya ΔChangeb Initialc Newd Initiale Newf Salmon
frequency –0.05 + 1.00 1.96 1.98
Average
quantity –0.18 + 3.60 421.78 436.96
Total 9.71 8.12
White fish
frequency –0.10 –2.00 3.77 3.69
Average quantity
–0.11 –2.20 435.03 425.46
Total 20.73 23.82
Total fish 30.44 31.94
Notes:
a The elasticity estimates are the estimates from Table VII.
b ΔChange is the percentage change in frequency or average quantity given a 20% price decrease for salmon and a 20% price increase for white fish.
c The initial frequencies and average quantities refer to the frequencies and average quantities as reported in Table II.
d The new frequency and average quantities are the new values after the 20% price decrease for salmon and the 20% price increase for white fish.
e Total revenue in € from sales of salmon and white fish given initial frequencies and quantities.
f Total revenue in €from sales of salmon and white fish given new frequencies and quantities.
32 Appendix 1. The Microeconomics Model
The Utility Maximization Problem
The consumer’s utility maximization problem is specified as:
max𝑛,𝑞,𝑙{𝑣(𝑥(𝑛𝑞), 𝑛, 𝑙): 𝑤𝑇 + 𝑅 = 𝑝′𝑛𝑞 + 𝑤𝑙 + 𝑤𝑔(𝑛), 𝑛 > 0, 𝑞 > 0, 𝑙 > 0} (1) with the associated Lagrangian problem:
ℒ = 𝑣(𝑥(𝑛𝑞), 𝑛, 𝑙) + 𝜆[𝑅 − 𝑝′𝑛𝑞 + 𝑤(𝑇 − 𝑙 − 𝑔(𝑛))]
The first-order conditions to this Lagrangian problem are:
𝜕ℒ
𝜕𝑞𝑖 = 𝜕𝑣
𝜕𝑥𝑖𝑛𝑖− 𝜆𝑝𝑖𝑛𝑖 = 0 (2)
𝜕ℒ
𝜕𝑛𝑖 = 𝜕𝑣
𝜕𝑥𝑖𝑞𝑖 + 𝜕𝑣
𝜕𝑛𝑖− 𝜆𝑝𝑖𝑞𝑖 − 𝜆𝑤 𝜕𝑔
𝜕𝑛𝑖 = 0 (3)
𝜕ℒ
𝜕𝑙 =𝜕𝑣
𝜕𝑙 − 𝜆𝑤 = 0 (4)
From Equations (2) and (4), we have:
𝜕𝑣
𝜕𝑥𝑖 1
𝑝𝑖 = 𝜆 (5)
𝜕𝑣
𝜕𝑙 1
𝑤 = 𝜆 (6)
By substituting for the first 𝜆 in Equation (3) by Equation (5), we get:
𝜕𝑣
𝜕𝑥𝑖𝑞𝑖+ 𝜕𝑣
𝜕𝑛𝑖 − 𝜕𝑣
𝜕𝑥𝑖 𝑝𝑖𝑞𝑖
𝑝𝑖 − 𝜆𝑤 𝜕𝑔
𝜕𝑛𝑖 = 0
𝜕𝑣
𝜕𝑥𝑖𝑞𝑖 + 𝜕𝑣
𝜕𝑛𝑖 − 𝜕𝑣
𝜕𝑥𝑖𝑞𝑖 − 𝜆𝑤 𝜕𝑔
𝜕𝑛𝑖 = 0
𝜕𝑣
𝜕𝑛𝑖 − 𝜆𝑤 𝜕𝑔
𝜕𝑛𝑖 = 0
33
𝜕𝑣/𝜕𝑛𝑖
𝜕𝑔/𝜕𝑛𝑖 = 𝜆𝑤 (7)
It follows from Equations (5), (6) and (7) that the necessary conditions for a maximum are:
𝜕𝑣/𝜕𝑥𝑖
𝜕𝑣/𝜕𝑥𝑗 = 𝑝𝑖
𝑝𝑗 ∀𝑖, 𝑗 (8)
𝜕𝑣/𝜕𝑙
𝜕𝑣/𝜕𝑞𝑖 = 𝑤
𝑝𝑖 ∀𝑖, 𝑗 (9)
𝜕𝑣/𝜕𝑛𝑖
𝜕𝑣/𝜕𝑙 = 𝜕𝑔
𝜕𝑛𝑖 ∀𝑖 (10)
𝜕𝑣/𝜕𝑛𝑖
𝜕𝑣/𝜕𝑛𝑗 =𝜕𝑔/𝜕𝑛𝑖
𝜕𝑔/𝜕𝑛𝑗 ∀𝑖, 𝑗 (11)
The solution of the first-order conditions is given by three sets of uncompensated demand functions: 𝑛(𝑝, 𝑤, 𝑅), 𝑞(𝑝, 𝑤, 𝑅) and 𝑙(𝑝, 𝑤, 𝑅). The total purchased quantities are given by 𝑛(𝑝, 𝑤, 𝑅) 𝑞(𝑝, 𝑤, 𝑅) = 𝑥(𝑝, 𝑤, 𝑅).
The Dual Cost Minimization Problem
The consumer’s cost minimization problem is given by:
min𝑛,𝑞,𝑙{𝑅 = 𝑝′𝑛𝑞 − 𝑤(𝑇 − 𝑙 − 𝑔(𝑛)): 𝑣(𝑥(𝑛𝑞), 𝑛, 𝑙) = 𝑣∗, 𝑛 > 0, 𝑞 > 0, 𝑙 > 0} (12) with the associated Lagrangian problem:
ℒ = 𝑝′𝑛𝑞 − 𝑤(𝑇 − 𝑙 − 𝑔(𝑛)) + 𝜆[𝑣∗− 𝑣(𝑥(𝑛𝑞), 𝑛, 𝑙)]
The first-order conditions to this Lagrangian problem are:
𝜕ℒ
𝜕𝑞𝑖 = 𝑝𝑖𝑛𝑖− 𝜆 𝜕𝑣
𝜕𝑥𝑖𝑛𝑖 = 0 (13)
𝜕ℒ
𝜕𝑛𝑖 = 𝑝𝑖𝑞𝑖 + 𝑤 𝜕𝑔
𝜕𝑛𝑖 − 𝜆𝜕𝑣
𝜕𝑥𝑖𝑞𝑖− 𝜆 𝜕𝑣
𝜕𝑛𝑖 = 0 (14)
34
𝜕ℒ
𝜕𝑙 = 𝑤 − 𝜆𝜕𝑣
𝜕𝑙 = 0 (15)
From Equations (13) and (15), we have Equations (5) and (6) above. By substituting for the first 𝜆 in Equation (14) by Equation (5), we get:
𝑝𝑖𝑞𝑖 + 𝑤𝜕𝑔
𝜕𝑛𝑖−
𝜕𝑣
𝜕𝑥𝑖
𝜕𝑣
𝜕𝑥𝑖
𝑝𝑖𝑞𝑖 − 𝜆 𝜕𝑣
𝜕𝑛𝑖 = 0
𝑝𝑖𝑞𝑖+ 𝑤 𝜕𝑔
𝜕𝑛𝑖 − 𝑝𝑖𝑞𝑖− 𝜆 𝜕𝑣
𝜕𝑛𝑖 = 0 𝑤 𝜕𝑔
𝜕𝑛𝑖 − 𝜆 𝜕𝑣
𝜕𝑛𝑖 = 0
𝜕𝑣/𝜕𝑛𝑖
𝜕𝑔/𝜕𝑛𝑖 =𝑤 𝜆
(16)
It follows from Equations (5), (6) and (16) that the necessary conditions for a minimum are given by Equations (8), (9), (10) and (11) above. The solution of the first-order conditions is three sets of compensated demand functions: 𝑛𝑐(𝑝, 𝑤, 𝑣∗), 𝑞𝑐(𝑝, 𝑤, 𝑣∗) and 𝑙𝑐(𝑝, 𝑤, 𝑣∗). The compensated demand function for total purchased quantities is 𝑛𝑐(𝑝, 𝑤, 𝑣∗)𝑞𝑐(𝑝, 𝑤, 𝑣∗) = 𝑥𝑐(𝑝, 𝑤, 𝑣∗).
Formulas for Compensated Price Elasticities
Under symmetry restrictions on a semi-log system, there are zero uncompensated cross- price elasticities; however, there are non-zero compensated cross-price elasticities that reflect the changes in total expenditures in the Slutsky equation. These compensated elasticities can be useful for calculating welfare effects in the model. The compensated price elasticity between products i and j for household k, 𝑒𝑖𝑗𝑘, are calculated as:
35 𝑒𝑖𝑗𝑘 = 𝑝𝑖𝑘𝑛𝑖𝑘
(𝑅𝑘+ 𝑦𝑘)
𝜕𝑛𝑗𝑘
𝜕(𝑅𝑘+ 𝑦𝑘)
(𝑅𝑘+ 𝑦𝑘)
𝑛𝑗𝑘 = 𝑠𝑛𝑖𝑘𝜔𝑛𝑗𝑘 (17)
where 𝑛𝑖𝑘 is the purchase frequency of product i by household k; 𝑠𝑛𝑖𝑘 is the total expenditure share of product i by household k, in terms of purchase frequencies; and 𝜔𝑛𝑗𝑘 is the total expenditure elasticity of product j by household k, in terms of purchase frequencies.
The compensated price elasticities for average purchased quantities are calculated as:
𝜀𝑖𝑗𝑘 = 𝑝𝑖𝑘𝑞𝑖𝑘 (𝑅𝑘+ 𝑦𝑘)
𝜕𝑞𝑗𝑘
𝜕(𝑅𝑘+ 𝑦𝑘)
(𝑅𝑘+ 𝑦𝑘)
𝑞𝑗𝑘 = 𝑠𝑞𝑖𝑘𝜔𝑞𝑗𝑘 (18)
where 𝑠𝑞𝑖𝑘 is the total expenditure share of product i by household k, in terms of
average quantities; and 𝜔𝑞𝑗𝑘 is the total expenditure elasticity of product j by household k, in terms of average quantities.
The estimated compensated elasticities for our model are provided in Table A1 at bottom of this Appendix. As we can see, compensated from the table, the cross-price effects are small.
Proposition 1.1
The demand equations 𝑛(𝑝, 𝑤, 𝑅), 𝑞(𝑝, 𝑤, 𝑅), 𝑙(𝑝, 𝑤, 𝑅) and 𝑥(𝑝, 𝑤, 𝑅) are homogeneous of degree zero in (𝑝, 𝑤, 𝑅).
Proof of Proposition 1.1
i) For the constraint of Equation (1) and any scalar 𝜌 > 0, we have:
{(𝑛, 𝑞, 𝑙)′ ∈ ℝ+2𝑀+1: 𝜌𝑅 ≥ 𝜌𝑝′𝑥 − 𝜌𝑤(𝑇 − 𝑙 − 𝑔(𝑛)), 𝑥 ≡ 𝑛𝑞}
